Dot Product and Orthogonality

$$\overline{u}\cdot\overline{v} = u_1v_1 + u_2v_2.$$

Show numbers works out the matching components multiplied and the products added. A component typed as a fraction, such as 1/2, stays a fraction in the worked lines.

With the sign regions on, a dashed line is drawn through the origin orthogonal to $\overline{u}$, and its two sides are shaded. Every $\overline{v}$ on the side that $\overline{u}$ points toward has a positive dot product with $\overline{u}$. Every $\overline{v}$ on the other side has a negative dot product. Every $\overline{v}$ on the dashed line has a dot product of 0: the vectors are orthogonal. A tip dragged close to the dashed line snaps onto it.

Length adds $\lVert\overline{u}\rVert = \sqrt{\overline{u}\cdot\overline{u}}$ and $\lVert\overline{v}\rVert = \sqrt{\overline{v}\cdot\overline{v}}$, worked out under Show numbers.

Components view has no picture and accepts vectors with any number of components. When the two vectors have different numbers of components, the dot product is marked undefined. The tool also shows the number that R’s dot() returns.

Under More:

• Order and scalars. The panel lists $\overline{v}\cdot\overline{u}$ beside $\overline{u}\cdot\overline{v}$. Scalar multiple adds $m\overline{u}$ and the check $(m\overline{u})\cdot\overline{v} = m(\overline{u}\cdot\overline{v})$.
• Board. With Snap off, dragged tips move in steps of 0.01.

Keys: press U or V to pick a tip. Press an arrow key to move it one grid step. With Snap off, the step is 0.1. Hold Shift with an arrow key to move it five steps.